How many ciritcal points functionf(x)=(x-1)|x-3|-4x+12 has? How many of these are points of local extrema?

Explanation

f(x) =(x-1)|x-3|-4x+12
`={9-x^(2), xlt3`
`x^(2)-8x+15,xge3`
`therefore f(x) ={-2x, xge3`
`22x-8,xgt3`
f(3)=-6,f(3)=-2
So f(x) is non differentiable at x=3
Also f(x) =0 and x=4
So , function has three critical points
Graph of the function is as shown in the following figure.

At x =3 derivation is not changing sign
So x =3 i snot a point of extrema.
At x = 0, derivation changes sign from positive to negative
So, x =0 0 is point of local maxima.
At x=3, derivative changes sign from negative to positive .
So , x=3 is point of local minima.